Summary: Concentration Profiles near an Activated Enzyme
Soohyung Park and Noam Agmon*
Institute of Chemistry and the Fritz Haber Research Center, The Hebrew UniVersity, Jerusalem 91904, Israel
ReceiVed: May 2, 2008; ReVised Manuscript ReceiVed: June 25, 2008
When a resting enzyme is activated, substrate concentration profile evolves in its vicinity, ultimately tending
to steady state. We use modern theories for many-body effects on diffusion-influenced reactions to derive
approximate analytical expressions for the steady-state profile and the Laplace transform of the transient
concentration profiles. These show excellent agreement with accurate many-particle Brownian-dynamics
simulations for the Michaelis-Menten kinetics. The steady-state profile has a hyperbolic dependence on the
distance of the substrate from the enzyme, albeit with a prefactor containing the complexity of the many-
body effects. These are most conspicuous for the substrate concentration at the surface of the enzyme. It
shows an interesting transition as a function of the enzyme turnover rate. When it is high, the contact
concentration decays monotonically to steady state. However, for slow turnover it is nonmonotonic, showing
a minimum due to reversible substrate binding, then a maximum due to diffusion of new substrate toward the
enzyme, and finally decay to steady state. Under certain conditions one can obtain a good estimate for the
critical value of the turnover rate constant at the transition.
I. Introduction
The Michaelis-Menten (MM) mechanism1 is a widely
applicable reaction scheme in biological processes. For enzy-
matic reactions, this mechanism has been routinely applied to